Quotient Rule for Logarithms
The quotient rule for logarithms states that the logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.
Formula
\log_b(x/y) = \log_b(x) - \log_b(y)
Definition
The quotient rule says the log of a fraction equals the log of the top minus the log of the bottom: for valid base $b$ and positive $x, y$, $\log_b(x/y) = \log_b(x) - \log_b(y)$. This follows because $x/y = b^m / b^n = b^{m-n}$ where $m = \log_b(x)$ and $n = \log_b(y)$, and is really just the product rule applied to the inverse: $\log(x/y) = \log(x \cdot y^{-1}) = \log(x) + \log(y^{-1}) = \log(x) - \log(y)$, so you only need to memorize the product rule. In harmonic analysis, the logarithmic derivative $d/dx[\ln(f(x))] = f'(x)/f(x)$ converts multiplicative changes in $f$ to additive ones, useful for analyzing products of functions.
Example
$\log_{10}(1000/10) = \log_{10}(1000) - \log_{10}(10) = 3 - 1 = 2$, matching $\log_{10}(100) = 2$. Condensing $\ln(x^2) - \ln(y^3) = \ln(x^2/y^3)$, and expanding $\log_5((x+1)/(x-2)) = \log_5(x+1) - \log_5(x-2)$ (which cannot simplify further since these are not a product or quotient of arguments). The logarithmic derivative of $f(x) = x^a \cdot e^{bx}$ is $f'(x)/f(x) = a/x + b$, decomposing the growth into a power-law component and an exponential component that can be identified separately.
Key Insight
Dividing inside a log becomes subtracting outside, mirroring how division and multiplication relate to subtraction and addition in the world of exponents. Logarithmic differentiation uses the quotient rule to simplify derivatives of complicated products and quotients: for $f = \text{numerator}/\text{denominator}$, $\ln|f| = \ln|\text{numerator}| - \ln|\text{denominator}|$, and differentiating gives a sum of simpler terms.